How long can I wait with blue skin tone to see if it goes away without seeing a doctor?

Depends. What do you mean by blue skin tone? If it is a local issue like a bruise, it will get better in a matter of days or weeks depending on the amount of blood in the tissue. If the bruise becomes painful, increases in size, becomes red or a discharge develops, see your doctor.

Related Questions

Do doctors see a lot of patients with blue skin tone?

Depends. Depends on the doctor and their patient load. I would imagine a pulmonologist, or lung doctor, probably sees more "blue" patients than a neurologist. Pathology also sees its fair share :).

I've been feeling more tired and have some blue skin tone. When should I see the doctor?

Fatigue. You are describing a host of possibilities although the most likely is anemia. A good physical exam and some laboratory work would be helpful from your family doctor. Sooner rather than later.
Yes... Hi eliza! Now, you must see a doc who understands and who knows to cure your illness.